On the local structure of the Euler-Lagrange mapping of the calculus of variations

TL;DR

This paper investigates the local structure of the Euler-Lagrange mapping in higher-order calculus of variations, providing new insights into its kernel, image, and conditions for PDE systems to be Euler-Lagrange equations.

Contribution

It offers a detailed description of the kernel and image of the higher-order Euler-Lagrange mapping and characterizes when PDE systems are derivable from variational principles.

Findings

Describes the kernel and image of the Euler-Lagrange mapping.

Provides explicit conditions for PDE systems to be Euler-Lagrange.

Advances understanding of the structure of variational calculus on fibered manifolds.

Abstract

The purpose of this paper is to announce some new results on the structure of the higher order Euler-Lagrange mapping of the multiple-integral variational calculus on fibered manifolds,namely a description of its kernel and its image,and an explicit characterization of the conditions under which a system of partial differential equations (of arbitrary order)is a system of the Euler--Lagrange equations.